complex sphere deformations
In the image above are three stages of the mesh. Left to Right. Hemisphere.  Hemisphere with a loop cut and delete vertices.  Last mesh image has subdivision surface modifier.




Create enough vertex density appropriate to the model.  Because there is a deep recess with edges near each other you should consider the object as possessing protrusions .. or arms.
Remove Vertices for flowing face density if possible
Hemisphere from sphere
Loop Cut for more density.  This prepares for the deep cut.  You can barely see how the Z height of the cut face area is now half of the surrounding face z height.  Thus allowing more detail.  Loop Cut can be searched.
Delete vertices on Both Sides.  This will allow the deep cut while keeping regular vertex density.
Consider Proportional edit or Lattice to shape the mesh. 
Subdivision Surface Modifier
Modify and Improve to your tastes
You are the Artist and can improve the crude mesh depicted above





Most forward mesh to viewer has used proportional edit.  Proportional edit may be searched here at BSE.